Lesson 1.1: Rational vs. Irrational Numbers & Decimal Expansions
Understand terminating, non-terminating recurring, and non-terminating non-recurring decimals.
⏱️ Reading Time: 18 mins
• ❓ 1 Interactive Practice Questions
Classification of Real Numbers
The set of Real Numbers (ℝ) is composed of two mutually exclusive subsets: Rational Numbers (ℚ) and Irrational Numbers.
1. Rational Numbers (ℚ)
A number is rational if it can be expressed in the form p / q where p and q are integers and q ≠ 0. Its decimal expansion is either:
- Terminating: Ends after a finite number of digits (e.g.,
3/8 = 0.375). - Non-terminating Recurring: A block of digits repeats infinitely (e.g.,
1/3 = 0.333...).
2. Irrational Numbers
Numbers that cannot be written as a simple fraction p / q. Their decimal expansions are non-terminating and non-recurring (e.g., √2 ≈ 1.4142135..., π ≈ 3.14159265...).
🎯 Key Learning Takeaways
• Every rational number has a terminating or repeating decimal expansion.
• Irrational numbers have non-terminating, non-repeating decimal expansions.
• Together, rational and irrational numbers complete the Real Number line.
✍️ Lesson Practice Questions (1 Questions)
Solve these real-world challenges online, test your understanding, and reveal instant step-by-step explanations.
1
Numerical
Medium
2 Marks
Convert the rational fraction 3 / 8 into its exact terminating decimal expansion:
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Correct Answer:
0.375 (±0.001)
Step-by-Step Explanation:
Dividing 3 by 8 gives 3 ÷ 8 = 0.375, which terminates after three decimal places.